3.42 \(\int \frac{x^6}{\cos ^{-1}(a x)} \, dx\)

Optimal. Leaf size=55 \[ -\frac{5 \text{Si}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac{9 \text{Si}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{5 \text{Si}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{\text{Si}\left (7 \cos ^{-1}(a x)\right )}{64 a^7} \]

[Out]

(-5*SinIntegral[ArcCos[a*x]])/(64*a^7) - (9*SinIntegral[3*ArcCos[a*x]])/(64*a^7) - (5*SinIntegral[5*ArcCos[a*x
]])/(64*a^7) - SinIntegral[7*ArcCos[a*x]]/(64*a^7)

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Rubi [A]  time = 0.0907464, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4636, 4406, 3299} \[ -\frac{5 \text{Si}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac{9 \text{Si}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{5 \text{Si}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{\text{Si}\left (7 \cos ^{-1}(a x)\right )}{64 a^7} \]

Antiderivative was successfully verified.

[In]

Int[x^6/ArcCos[a*x],x]

[Out]

(-5*SinIntegral[ArcCos[a*x]])/(64*a^7) - (9*SinIntegral[3*ArcCos[a*x]])/(64*a^7) - (5*SinIntegral[5*ArcCos[a*x
]])/(64*a^7) - SinIntegral[7*ArcCos[a*x]]/(64*a^7)

Rule 4636

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Cos[x]^m*Sin[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x^6}{\cos ^{-1}(a x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^6(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^7}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{5 \sin (x)}{64 x}+\frac{9 \sin (3 x)}{64 x}+\frac{5 \sin (5 x)}{64 x}+\frac{\sin (7 x)}{64 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^7}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (7 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac{5 \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac{5 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac{9 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}\\ &=-\frac{5 \text{Si}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac{9 \text{Si}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{5 \text{Si}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{\text{Si}\left (7 \cos ^{-1}(a x)\right )}{64 a^7}\\ \end{align*}

Mathematica [A]  time = 0.102637, size = 40, normalized size = 0.73 \[ -\frac{5 \text{Si}\left (\cos ^{-1}(a x)\right )+9 \text{Si}\left (3 \cos ^{-1}(a x)\right )+5 \text{Si}\left (5 \cos ^{-1}(a x)\right )+\text{Si}\left (7 \cos ^{-1}(a x)\right )}{64 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/ArcCos[a*x],x]

[Out]

-(5*SinIntegral[ArcCos[a*x]] + 9*SinIntegral[3*ArcCos[a*x]] + 5*SinIntegral[5*ArcCos[a*x]] + SinIntegral[7*Arc
Cos[a*x]])/(64*a^7)

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Maple [A]  time = 0.058, size = 40, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{7}} \left ( -{\frac{9\,{\it Si} \left ( 3\,\arccos \left ( ax \right ) \right ) }{64}}-{\frac{5\,{\it Si} \left ( 5\,\arccos \left ( ax \right ) \right ) }{64}}-{\frac{{\it Si} \left ( 7\,\arccos \left ( ax \right ) \right ) }{64}}-{\frac{5\,{\it Si} \left ( \arccos \left ( ax \right ) \right ) }{64}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/arccos(a*x),x)

[Out]

1/a^7*(-9/64*Si(3*arccos(a*x))-5/64*Si(5*arccos(a*x))-1/64*Si(7*arccos(a*x))-5/64*Si(arccos(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\arccos \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arccos(a*x),x, algorithm="maxima")

[Out]

integrate(x^6/arccos(a*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{6}}{\arccos \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arccos(a*x),x, algorithm="fricas")

[Out]

integral(x^6/arccos(a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\operatorname{acos}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/acos(a*x),x)

[Out]

Integral(x**6/acos(a*x), x)

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Giac [A]  time = 1.15947, size = 63, normalized size = 1.15 \begin{align*} -\frac{\operatorname{Si}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac{5 \, \operatorname{Si}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac{9 \, \operatorname{Si}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac{5 \, \operatorname{Si}\left (\arccos \left (a x\right )\right )}{64 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/arccos(a*x),x, algorithm="giac")

[Out]

-1/64*sin_integral(7*arccos(a*x))/a^7 - 5/64*sin_integral(5*arccos(a*x))/a^7 - 9/64*sin_integral(3*arccos(a*x)
)/a^7 - 5/64*sin_integral(arccos(a*x))/a^7